The subject of this paper involves properties of composition operators
on holomorphic function spaces on the right half-plane \({\mathbb {C}}_+\) , both
as individual operators and as elements of one-parameter semigroups.
One difficulty, even in the case of the Hardy space \(H^2({\mathbb {C}}_+)\) , is that not all
composition operators on the spaces are bounded, although in many cases the
bounded operators have been characterised, as we explain below.
The literature on semigroups of composition operators is not as extensive as it is for the
disc, although it has been an object of study since the work of Berkson and Porta [1]}.
